Math 234 — Lecture Schedule
We will cover the material in the text in the following order and pace. Since
there are 43 lectures, and 30 items below, each item should take a bit longer
than one lecture, leaving time for review, three midterms, question&answer, etc.
- Chapter 1.
Vectors : dot product, (cross product), defining equations for lines and
planes.
This should be a quick introduction; the cross product will be
needed occasionally, but the dot-product and defining equations for
lines and planes do show up a lot.
- Ch.2, §1–9.
Parametric curves: lines, circles, helix; velocity & acceleration.
- Ch.2 §10–16.
Parametric curves: arc length, unit tangent, curvature, normal for
space curves, the osculating plane.
- Ch.3 §1–5.
Functions of several variables: level sets, linear functions,
quadratic forms (completing the square to classify them), *functions
in polar coordinates (the helicoid)
- Ch.4 §1–5.
Derivatives: partials, the linear approximation, tangent plane to a
graph
- Ch.4 §6–9.
The Chain Rule and the gradient of a function
- *Ch.4 §10.
Implicit function theorem
- Ch.4 §11.
The Chain Rule with more variables: coordinate transformations.
- Ch.4 §13–14.
Higher partial derivatives and “Clairaut’s theorem”
(mixed partials are equal)
-
Special lecture: the wave equation.
- Ch.5 §1–6.
Define Maxima, Minima, discuss “continuous functions on closed
and bounded sets,” explain why interior maxima and minima are
critical points (but not always the other way around).
- Ch.5 §7.
Special lecture on linear regression as an example.
- Ch.5 §9.1–9.5.
Discuss Taylor expansion of order 2, as preparation for the second
derivative test.
- Ch.5 §9.5–9.7, §11.
Second derivative test, examples; for saddle points show how you get
the tangents to the two branches.
- Ch.5 §12.
Lagrange multipliers
- Ch.6 §1–2.4.
Integration: the general idea, double integral over a rectangle can be
computed as an iterated integral.
- Ch.6 §2.5–3.
Double integral examples, and double integral over non-rectangles.
- Ch.6 §4–5.3.
An example of switching the order of integration. Double integral in
polar coordinates.
- Ch.6 §5.4–5.8.
Triple integrals, definition, expression in terms of iterated
integrals, examples
- Ch.6 §6
Applications of triple (and double) integrals. Averages of functions,
the concept of a density, moment of inertia.
- Ch.6 §1–2.4.
Triple integrals in special coordinate systems: spherical and
cylindrical coordinates. Derive the “volume element” and show
some examples.
- Ch7 §3
Start vector calculus; define line integrals of functions
(“scalars”)
- Ch.7 §1, 2, §5.1
Describe vector fields, line integral of a vector field,
interpretation as “work”
- Ch.7 §5
Examples of line integrals around closed curves, along piecewise
defined curves.
- Ch.7 §6, 7
Fundamental Theorem: line integral of a gradient does not depend on
the path, definition of a conservative vector field
- Ch.7 §9
Flux integrals (for fluid flows in the plane); example, flux through a
closed curve and the “production of stuff” (leads to the idea
that the integral over the boundary of a region may have something to do
with what happens inside the region).
- Ch.7 §10
Green’s theorem—both line integral and flux versions; examples
(e.g. the expanding flow again)
- Ch.7 §11
Conservative fields and Green’s theorem; the problem with
non-simply connected domains.
- Ch.7 §13
Surfaces and Surface Integrals; flux through a surface.
- Ch.7 §14, 15
Divergence theorem and Stokes’ theorem.
- Ch.7 §16
“∇”, ∇·f,
;∇×f, divergence of curl is zero, curl of gradient is
zero.